Do pseudoforces arise due to inertia? Wikipedia says

The fictitious force is due to an object's inertia when the reference frame does not move inertially, and thus begins to accelerate relative to the free object. The fictitious force thus does not arise from any physical interaction between two objects, such as electromagnetism or contact forces, but rather from the acceleration a of the non-inertial reference frame itself, which from the viewpoint of the frame now appears to be an acceleration of the object instead, requiring a "force" to make this happen.

The first line says pseudoforces arise due to inertia. The very next line says that they arise due to acceleration of the reference frame itself.
I agree with the second point. However, the first point is a bit confusing.
The first point seems reasonable when describing the pseudoforce on the upper part of our body in a bus when the bus suddenly stops or starts.
However, when describing pseudoforces like centrifugal force, it makes no sense to me. Can anyone explain how is it so?
EDIT
The way I phrase my question is somewhat ambiguous. Here's a more clearer one.
My question is not about centrifugal force, my question is how centrifugal force can be explained as a result of inertia.
I meant it to be the first point, not the concept.
 A: In the inertial frame (not the accelerating frame):

*

*When standing in a bus on roller skates, when the bus brakes you will move forwards. As you described yourself, it seems as though you are being pushed forward with some non-existing centrifugal "force". But in reality there is no such force - it is just the bus which is being pulled backwards from underneath your feet. You had forwards speed before the bus started braking, and due to the inertia of your body, your body tends to keep that forwards speed while the bus slows down below your feet.

Now we'll employ this same idea to circular motion (or any turning motion):

*

*When sittin in a car your body is in motion forwards. Then the car turns. You feel squeezed against the car door as if a force is pushing you outwards. But in reality it is just the car which is moving into you. Due to inertia of your body, your body tends to keep moving straight ahead, so when the car turns that means that the seat underneath you moves away from underneath you and the car door comes slamming into you.
If the car continues turning so it forms a circular motion, then this experience happens constantly and you will constantly feel squeezed against the car door. But it is rather the car door which is constant changing directly and moving into you and pulling you along.

The principle of inertia causing the tendency to continue moving with constant speed and direction applies in both cases.
A: 
The first point seems reasonable when describing the pseudoforce on
the upper part of our body in a bus when the bus suddenly stops or
starts.
However, when describing pseudoforces like centrifugal force, it makes
no sense to me. Can anyone explain how is it so?

The first example, where the bus suddenly stops or starts, describes a force causing a change in the magnitude of linear velocity (the speed of an object) in a non-inertial frame, i.e., acceleration that changes the magnitude of the velocity of an object in rectilinear motion but not its direction.
The centrifugal force describes the force that arises from a change in the direction of the velocity of an object but not its magnitude (speed). It is the force that acts outward on a body moving around a center due to its inertia, such as the apparent force that pushes you to one side of the bus when it suddenly turns.
Both are fictitious forces, or inertial forces, that rather arise from the acceleration of the non-inertial reference frame itself. Fictitious forces are needed in a non-inertial frame to enable the application Newton's laws of motion.

I guess you misunderstood. What I asked was relation between
pseudoforces and inertia.

It seemed to me you wanted to know the difference between the stop and start bus force and centrifugal force. In any case, a pseudo force is also called a fictitious or inertial force. It is used to explain the the apparent acceleration of an object in a non-inertial frame.

So, why do we call it inertial forces?

Because a body at rest or in uniform rectilinear motion in an inertial (non accelerating) frame looks like it is accelerating when observed in a non-inertial (accelerating) frame. To explain the acceleration when observed in the non-inertial (accelerating) frame we introduce the term "inertial force", "pseudo force" or "fictitious force".
Hope this helps.
A: Fictitious forces arise in an accelerating reference frame. The different types of fictitious forces- centrifugal, Coriolis, etc.- appear when modifying Newton's second law considering this acceleration; see a physics mechanics textbook for the development, such as Symon Mechanics.
Inertia is the property of a body that resists acceleration.  In a non-accelerating reference frame a true force is required to accelerate an object due to its inertia, so a non-accelerating reference frame is called an inertial frame. In an accelerating reference frame the object is subject to additional fictitious forces that do not appear in the non-accelerating (inertial) frame, hence this accelerating reference frame is called a non-inertial frame.
Regarding your specific question in your edited question. Inertia is described by mass.  All the fictitious forces depend on the mass (see Symon, Mechanics); hence they are called inertial forces, and their magnitude depends on mass (inertia).  A fictitious force is explained by the motion of an object in an accelerating reference frame; a fictitious force is linearly dependent on inertia (mass).
A: 
My question is not about centrifugal force, my question is how centrifugal force can be explained as a result of inertia.

Inertia is related to Newton’s first and second laws.
The first law implies that non-interacting objects travel in a straight line at a constant speed. Such an object is said to be moving under its own inertia. This can be considered as an operational definition of an inertial frame.
The second law describes the force required to make an object deviate from the “straight line constant speed” inertial path. This can be considered as an operational definition of force.
Now, if we have a non-interacting object then, by definition, it travels in a straight line at a constant speed in an inertial frame. Again, this is motion under its own inertia. If we transform the coordinates to non-inertial coordinates then we will find that the non-interacting object follows a path that is not a straight line in those coordinates. In other words, its motion under inertia has coordinate acceleration.
We can then use the second law to determine the force that would be required to produce such an acceleration in inertial coordinates. This force is not due to any interaction (the object is non-interacting), but simply due to the object’s motion under inertia as described in the non-inertial frame. Hence it is called an inertial force or more commonly a fictitious force.
A: "Inertia" really refers to the relationship between force, mass, and acceleration; and roughly equates to mass per se.  "Fictitious" forces like centrifugal force (and gravity, in the context of general relativity) produce equal acceleration for all objects regardless of charge, mass, etc.  So, IMHO, inertia - which is essentially an aspect of mass - should not be invoked as an explanation of centrifugal force.
On the other hand, it takes a real force to oppose a fictitious force such as gravity or centrifugal force.  If you are in a carousel and you feel centrifugal force, what you're actually feeling is the real force that is opposing centrifugal force: you are feeling a centripetal force.
